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Unit Roots and Cointegration in Panels

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Unit Roots and Cointegration in Panels UNIT ROOTS AND COINTEGRATION IN PANELS JOERG BREITUNG M. HASHEM PESARAN CESIFO WORKING PAPER NO. 1565 CATEGORY 10: EMPIRICAL AND THEORETICAL METHODS OCTOBER 2005 An electronic version of the paper may be downloaded • from the SSRN website: www.SSRN.com • from the CESifo website: www.CESifo-group.de CESifo Working Paper No. 1565 UNIT ROOTS AND COINTEGRATION IN PANELS Abstract This paper provides a review of the literature on unit roots and cointegration in panels where the time dimension (T) and the cross section dimension (N) are relatively large. It distinguishes between the first generation tests developed on the assumption of the cross section independence, and the second generation tests that allow, in a variety of forms and degrees, the dependence that might prevail across the different units in the panel. In the analysis of cointegration the hypothesis testing and estimation problems are further complicated by the possibility of cross section cointegration which could arise if the unit roots in the different cross section units are due to common random walk components. JEL Code: C12, C15, C22, C23. Keywords: panel unit roots, panel cointegration, cross section dependence, common effects. Joerg Breitung M. Hashem Pesaran University of Bonn Cambridge University Department of Economics Sidgwick Avenue Institute of Econometrics Cambridge, CB3 9DD Adenauerallee 24 – 42 United Kingdom 53113 Bonn [email protected] Germany [email protected] We are grateful to Jushan Bai, Badi Baltagi, George Kapetanios, Uwe Hassler, Serena Ng, Elisa Tosetti, Ron Smith, and Joakim Westerlund for comments on a preliminary version of this paper. 1 Introduction Recent advances in time series econometrics and panel data analysis have focussed attention on unit root and cointegration properties of variables ob- served over a relatively long span of time across a large number of cross sec- tion units, such as countries, regions, companies or even households. Such panel data sets have been used predominately in testing the purchasing power parity and output convergence, although the panel techniques have also been adapted more recently to the analysis of business cycle synchronization, house price convergence, regional migration and household income dynamics. This paper provides a review of the theoretical literature on testing for unit roots and cointegration in panels where the time dimension (T ), and the cross sec- tion dimension (N) are relatively large. In cases where N is large (say over 100) and T small (less than 10) the analysis can proceed only under restric- tive assumptions such as dynamic homogeneity and/or local cross section dependence as in spatial autoregressive or moving average models. In cases where N is small (less than 10) and T is relatively large standard time series techniques applied to systems of equations, such as the Seemingly Unrelated Regression Equations (SURE), can be used and the panel aspect of the data should not pose new technical di¢ culties. One of the primary reasons behind the application of unit root and coin- tegration tests to a panel of cross section units was to gain statistical power and to improve on the poor power of their univariate counterparts. This was supported by the application of what might be called the …rst genera- tion panel unit root tests to real exchange rates, output and in‡ation. For example, the augmented Dickey-Fuller test is typically not able to reject the hypothesis that the real exchange rate is nonstationary. In contrast, panel unit root tests applied to a collection of industrialized countries generally …nd that real exchange rates are stationary, thereby lending empirical sup- port to the purchasing power parity hypothesis (e.g. Coakley and Fuertes (1997) and Choi (2001)). Unfortunately, testing the unit root and cointegration hypotheses by us- ing panel data instead of individual time series involves several additional complications. First, panel data generally introduce a substantial amount of unobserved heterogeneity, rendering the parameters of the model cross section speci…c. Second, in many empirical applications, particularly the application to the real exchange rates mentioned above, it is inappropriate to assume that the cross section units are independent. To overcome these di¢ culties, variants of panel unit root tests are developed that allow for dif- 2 ferent forms of cross sectional dependence.1 Third, the panel test outcomes are often di¢ cult to interpret if the null of the unit root or cointegration is rejected. The best that can be concluded is that "a signi…cant fraction of the cross section units is stationary or cointegrated". The panel tests do not provide explicit guidance as to the size of this fraction or the identity of the cross section units that are stationary or cointegrated. Fourth, with unob- served I(1) (i.e. integrated of order unity) common factors a¤ecting some or all the variables in the panel, it is also necessary to consider the possi- bility of cointegration between the variables across the groups (cross section cointegration) as well as within group cointegration. Finally, the asymptotic theory is considerably more complicated due to the fact that the sampling design involves a time as well as a cross section dimension. For example, applying the usual Dickey-Fuller test to a panel data set introduces a bias that is not present in the case of a univariate test. Furthermore, a proper limit theory has to take into account the relationship between the increasing number of time periods and cross section units (cf. Phillips and Moon 1999). By comparison to panel unit root tests, the analysis of cointegration in panels is still at an early stages of its developments. So far the focus of the panel cointegration literature has been on residual based approaches, al- though there has been a number of attempts at the development of system approaches as well. As in the case of panel unit root tests, such tests are developed based on homogenous and heterogeneous alternatives. The resid- ual based tests were developed to ward against the "spurious regression" problem that can also arise in panels when dealing with I(1) variables. Such tests are appropriate when it is known a priori that at most there can be only one within group cointegration in the panel. System approaches are required in more general settings where more than one within group coin- tegrating relation might be present, and/or there exist unobserved common I(1) factors. Having established a cointegration relationship, the long-run parameters can be estimated e¢ ciently using techniques similar to the ones proposed in the case of single time series models. Speci…cally, fully-modi…ed OLS pro- cedures, the dynamic OLS estimator and estimators based on a vector error correction representation were adopted to panel data structures. Most ap- proaches employ a homogenous framework, that is, the cointegration vectors are assumed to be identical for all panel units, whereas the short-run pa- rameters are panel speci…c. Although such an assumption seems plausible 1 In fact the application of the second generation panel unit root tests to real exchange rates tend to over-turn the earlier test results that assume the cross section units are independently distributed. See Moon and Perron (2004) and Pesaran (2005). 3 for some economic relationships (like the PPP hypothesis mentioned above) there are other behavioral relationships (like the consumption function or money demand), where a homogeneous framework seems overly restrictive. On the other hand, allowing all parameters to be individual speci…c would substantially reduce the appeal of a panel data study. It is therefore impor- tant to identify parameters that are likely to be similar across panel units whilst at the same time allowing for su¢ cient heterogeneity of other para- meters. This requires the development of appropriate techniques for testing the homogeneity of a sub-set of parameters across the cross section units. When N is small relative to T; standard likelihood ratio based statistics can be used. Groen and Kleibergen (2003) provide an application. Testing for parameter homogeneity in the case of large panels poses new challenges that require further research. Some initial attempts are made in Pesaran, Smith and Im (1996), Phillips and Sul (2003a) and Pesaran and Yamagata (2005). This paper reviews some recent work in this rapidly developing research area and thereby updating the earlier excellent surveys of Banerjee (1999), Baltagi and Kao (2000) and Choi (2004). The remainder of the paper is or- ganized as follows: Section 2 sets out the basic model for the panel unit root tests and describes the …rst generation panel unit root tests. Second genera- tion panel unit root tests are described in Section 3, and a brief account of the small sample properties of the panel unit root tests is provided in Section 4. General issues surrounding panel cointegration, including the problem of cross-section cointegration, are discussed in Section 5. Residual-based and system approaches to testing for cointegration in panels are reviewed in Sections 6 and 7; and estimation of the cointegration relations in panels is discussed in Section 8. Panels with unobserved common factors, allowing for cross-section cointegration, are reviewed in Section 9. Some concluding remarks are provided in Section 10. 2 First Generation Panel Unit Root Tests 2.1 The Basic Model Assume that time series yi0; : : : ; yiT on the cross section units i = 1; 2; :::; N are generated for each i byf a simple …rst-orderg autoregressive, AR(1), process yit = (1 i)i + iyi;t 1 + "it; (1) where the initial values, yi0, are given, and the errors "it are identically, 2 independently distributed (i.i.d.) across i and t with E("it) = 0, E("it) = 2 < and E("4 ) < . These processes can also be written equivalently i 1 it 1 4 as simple Dickey-Fuller (DF) regressions yit = ii + iyi;t 1 + "it ; (2) where yit = yit yi;t 1, i = i 1.
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